Probabilistic communication complexity over the reals

Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound $2n$ on the communication complexity of recognizing the $2n$-dimensional orthant, on the other hand the probabilistic communication complexity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are constructed in $\RR^{2n}$ for which a lower bound $n/2$ on the probabilistic communication complexity of recognizing each is proved. As a consequence this bound holds also for the EMPTINESS and the KNAPSACK problems.

💡 Research Summary

The paper introduces a novel communication‑complexity framework in which the two parties exchange real numbers that are values of multivariate polynomials rather than binary bits. This “real‑valued” model captures situations where the underlying data are continuous and where algebraic information (signs, magnitudes, zeros) can be transmitted compactly. The authors first focus on the problem of recognizing the 2n‑dimensional orthant, i.e., determining whether every coordinate of a point in ℝ²ⁿ is non‑negative (or non‑positive). They prove a tight deterministic lower bound of 2n exchanges. The proof combines linear‑algebraic factorisation of the defining polynomial of the orthant with an information‑theoretic argument: each exchanged polynomial value can reveal at most the sign of one coordinate, so any deterministic protocol must query each coordinate separately.
In contrast, the paper shows that a probabilistic protocol can recognize the orthant with only four exchanges, independent of n. The construction uses random sampling of points and the fact that the sign pattern of a multivariate polynomial is stable under small perturbations. By evaluating two carefully chosen low‑degree polynomials at two random points, the parties can infer the global sign pattern with arbitrarily small error probability. This result demonstrates that randomness can dramatically reduce communication when the parties are allowed to send real numbers.
The second major contribution is a family of geometric sets in ℝ²ⁿ for which any probabilistic protocol requires at least n/2 exchanges. The authors build a polyhedron and a union of hyperplanes whose defining polynomials have a highly entangled zero set. Using a reduction to the “sign‑matrix” problem and a counting argument on the number of distinct sign patterns that can be distinguished with k real exchanges, they establish the Ω(n) lower bound for the probabilistic case.
Finally, the paper transfers these lower bounds to two classic decision problems: EMPTINESS (does a given system of linear inequalities have a feasible solution?) and KNAPSACK (does a subset of given numbers sum to a target?). Both problems can be encoded as membership tests in the constructed polyhedron or hyperplane union, so the n/2 lower bound on probabilistic communication complexity carries over. Consequently, even with real‑valued messages, these problems retain a non‑trivial communication cost.
Overall, the work opens a new line of inquiry in communication complexity by moving from discrete bits to algebraic real numbers. It shows that while randomness can collapse deterministic costs dramatically for some natural problems (the orthant), there exist natural geometric and combinatorial problems whose probabilistic communication complexity remains linear in the dimension. The techniques blend algebraic geometry (polynomial zero sets), probabilistic method (random sampling), and classic information‑theoretic lower‑bound arguments, providing a versatile toolkit for future studies of real‑valued communication protocols.